A central question in community ecology is how the number of trophic links relates to community species richness. For simple dynamical food-web models, link density (the ratio of links to species) is bounded from above as the number of species increases; but empirical data suggest that it increases without bounds. We found a new empirical upper bound on link density in large marine communities with emphasis on fish and squid, using novel methods that avoid known sources of bias in traditional approaches. Bounds are expressed in terms of the diet-partitioning function (DPF): the average number of resources contributing more than a fraction f to a consumer's diet, as a function of f. All observed DPF follow a functional form closely related to a power law, with power-law exponents independent of species richness at the measurement accuracy. Results imply universal upper bounds on link density across the oceans. However, the inherently scale-free nature of power-law diet partitioning suggests that the DPF itself is a better defined characterization of network structure than link density.